1 Class: FY BSc Subject : Calculus Chapter: Unit 2 Chapter 3 Chapter Name: Derivatives 1 Today’s Agenda 2 Derivatives Derivatives Notation Why to study derivatives? Application of Derivatives in daily life Differentiation Formulas Higher Order Derivatives Differentiation Rules Chain Rule of Differentiation Partial Differentiation Derivative 3 Derivative, in mathematics is defined as the rate of change of a function with respect to a variable. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. 1 https://www.youtube.com/watch?v=rAof9Ld5sOg Let f(x) be a function defined in an open interval containing a. The derivative of the function f(x) at a, denoted by f′(a), is defined by : f’(a)= 3 Derivatives notation 4 1.1 Derivatives are the result of performing a differentiation process upon a function or an expression. Derivative notation is the way we express derivatives mathematically. 1. Lagrange's notation : In Lagrange's notation, the derivative of f is expressed as f' prime (pronounced "f prime" ). This notation is probably the most common when dealing with functions with a single variable. 2. Leibniz's notation In Leibniz's notation, the derivative of f is expressed as  This notation, becomes very useful when dealing with integral calculus, differential equations, and multivariable calculus. 3. Newton's notation In Newton's notation, the derivative of f is expressed as   and the derivative of  y= f(x) is expressed as This notation is mostly common in Physics and other sciences where calculus is applied in a real-world context. 4 Why to study derivatives ? 5 1.2 The original purpose of the derivative is to analyze the sensitivity or rate of change of a function with respect to its independent variable — that is, given a tiny change in the independent variable x, how much does the dependent variable, y, respond to that change. If the derivative of a function is high, it means that, for that function, the dependent variable responds greatly to that tiny change in its independent variable; if the derivative is low, the dependent variable changes a very small amount in response. When taking the derivative of a function, you can find relative extrema, critical values, and where the function is increasing decreasing. Taking the derivative again, you can find the concavity of the function, and the points of inflection that exist on it.  Derivatives are used in velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. 5 Application of derivatives in daily life 6 1.3 The derivative can be used for many other purposes, such as analyzing the increasing/decreasing behavior and concavity of a function. Example: Let’s say the population in an island at a certain time t can be modeled by the function P(t). If, say, the population is increasing, we know that after a tiny change in time can result in a tiny, positive change in population; if the population is decreasing, a tiny change in time results in a tiny, negative change in population. Using what we discussed above, the derivative of P, can tell us the increasing / decreasing behavior of population with its sign. If   is positive, it means that a tiny change in time is accompanied by a tiny positive growth in population; the same logic goes with negative. Hence, if the derivative is positive, the original function is increasing; if the derivative is negative, the original function is decreasing. 6 Differentiation Formulas 7 2 1. 2. 3. 7 Example 8 2 Example 1: = 0 Example 2: = 5(1) +0 = 5 8 Differentiation Formulas 9 2 4. 5. 9 Example 10 2 Example 1: = 8x7 Example 2: = 6(-4) x-4-1 = -24 x-5 10 Differentiation Formulas 11 2 6. 7. 8. 11 Question 12 Find the derivative of the following functions: f(x)= x3 + 9x2 -48x +2 f(x) = y-4 -9y-3 +8y-2 +12 12 Solution 13 f’(x) = 3x2 +18x -48 = 3(x2 +6x -16) = 3(x+8)(x−2) f’(x) = -4y-4-1-9(-3)y-3-1 +8(-2)y-2-1+0 = -4y-5 +27y-4 -16y-3 13 Example 14 2 1. A toy company can sell x electronic gaming systems at a price of p=−0.01x+400 dollars per gaming system .The cost of manufacturing x systems is given by C(x)=100x+10,000 dollars .Find the rate of change of profit when 10,000 games are produced. Should the toy company increase or decrease production? Solution : The profit P(x) earned by producing x gaming systems is R(x)−C(x), Where, R(x) is the revenue obtained from the sale of x games. Since the company can sell x games at p=−0.01x+400 per game, R(x)= x p = x(−0.01x+400) =−0.01 x2+400x. Consequently, P(x)=−0.01 x2+300x−10,000. 14 Example 15 2 Therefore, evaluating the rate of change of profit gives : Since the rate of change of profit P′(10,000)>0 and P(10,000)>0, the company should increase production. 15 Higher-Order Derivatives 16 3 The second derivative of a function is just the derivative of its first derivative. The third derivative is the derivative of the second derivative, the fourth derivative is the derivative of the third, and so on The derivative of f′( x), which is generally referred to as the second derivative of f(x) and written f“( x) or f 2( x). For example, The derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. 16 Example 17 3  1. f(x) = x3 Solution: First derivative is  f'(x) = 3x2 Second derivative is f’’(x) = 6x 17 Question 18 f(x) = (4x2 +1)2 Find the second derivative of the following function: 1. 18 Solution 19 19 Differentiation rules 20 4 1. Sum rule: f(x)=g(x)+h(x) Then, f'(x)=g'(x)+h'(x). 20 Example 21 4 1.f(x) = x + x3 Solution: f’(x) = u’(x) + v’(x) Now, differentiating the given function, we get; f’(x) = d/dx (x + x3) f’(x) = d/dx (x) + d/dx (x3) f’(x) = 1 + 3x2 21 Differentiation rules 22 4 2.Product Rule: f(x) = u(x)×v(x), then: f′(x) = u′(x) × v(x) + u(x) × v′(x) 22 Example 23 4 1.f(x)=x2(x+3) Solution: f′(x) = u′(x) × v(x) + u(x) × v′(x) Here, u(x) = x2 and v(x) = x+3 Therefore, on differentiating the given function, we get; f’(x) = d/dx [x2(x+3)] f’(x) = d/dx (x2)(x+3)+x2 d/dx (x+3) f’(x) = 2x(x+3)+x2(1) f’(x) = 2x2+6x+x2 f’(x) = 3x2+6x f’(x) = 3x(x+2)   23 Differentiation rules 24 4 3. Quotient Rule: f(x) = u(x)/v(x) 24 Example 25 4 1. Solution: 25 Chain Rule of Differentiation 26 5 If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation is defined as; If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation is defined as; 26 Example 27 5 1. Solution: = 27 Question 28 Solve the following function and find its first derivative: 1. 28 Solution 29 29 Partial Differentiation 30 6 The process of finding the partial derivative of a function is called partial differentiation. In this process, the partial derivative of a function with respect to one variable is found by keeping the other variable constant. A partial derivative is defined as a derivative in which some variables are kept constant and the derivative of a function with respect to the other variable can be determined. The partial derivative of a function “f’ with respect to “x” is represented by f x or ∂f/∂x. The partial derivative of a function “f’ with respect to “y” is represented by f y or ∂f/∂y. 30 Example 31 6 Determine the partial derivative of the function: f (x,y) = 3x + 4y. Solution: Given function: f (x,y) = 3x + 4y To find ∂f/∂x, keep y as constant and differentiate the function: Therefore, ∂f/∂x = 3 Similarly, to find  ∂f/∂y, keep x as constant and differentiate the function: Therefore, ∂f/∂y = 4 31 Question 32 Find the derivative of the following function with respect to x and y: 1. f(x, y) = x4 y3 +8x2 y +y4 +5x 32 Solution 33 33